4,049 research outputs found
Wavefunction localization and its semiclassical description in a 3-dimensional system with mixed classical dynamics
We discuss the localization of wavefunctions along planes containing the
shortest periodic orbits in a three-dimensional billiard system with axial
symmetry. This model mimicks the self-consistent mean field of a heavy nucleus
at deformations that occur characteristically during the fission process [1,2].
Many actinide nuclei become unstable against left-right asymmetric
deformations, which results in asymmetric fragment mass distributions. Recently
we have shown [3,4] that the onset of this asymmetry can be explained in the
semiclassical periodic orbit theory by a few short periodic orbits lying in
planes perpendicular to the symmetry axis. Presently we show that these orbits
are surrounded by small islands of stability in an otherwise chaotic phase
space, and that the wavefunctions of the diabatic quantum states that are most
sensitive to the left-right asymmetry have their extrema in the same planes. An
EBK quantization of the classical motion near these planes reproduces the exact
eigenenergies of the diabatic quantum states surprisingly well.Comment: 4 pages, 5 figures, contribution to the Nobel Symposium on Quantum
Chao
Semiclassical Theory of Chaotic Quantum Transport
We present a refined semiclassical approach to the Landauer conductance and
Kubo conductivity of clean chaotic mesoscopic systems. We demonstrate for
systems with uniformly hyperbolic dynamics that including off-diagonal
contributions to double sums over classical paths gives a weak-localization
correction in quantitative agreement with results from random matrix theory. We
further discuss the magnetic field dependence. This semiclassical treatment
accounts for current conservation.Comment: 4 pages, 1 figur
Semiclassical universality of parametric spectral correlations
We consider quantum systems with a chaotic classical limit that depend on an
external parameter, and study correlations between the spectra at different
parameter values. In particular, we consider the parametric spectral form
factor which depends on a scaled parameter difference . For
parameter variations that do not change the symmetry of the system we show by
using semiclassical periodic orbit expansions that the small expansion
of the form factor agrees with Random Matrix Theory for systems with and
without time reversal symmetry.Comment: 18 pages, no figure
Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs
Physicists have argued that periodic orbit bunching leads to universal
spectral fluctuations for chaotic quantum systems. To establish a more detailed
mathematical understanding of this fact, it is first necessary to look more
closely at the classical side of the problem and determine orbit pairs
consisting of orbits which have similar actions. In this paper we specialize to
the geodesic flow on compact factors of the hyperbolic plane as a classical
chaotic system. We prove the existence of a periodic partner orbit for a given
periodic orbit which has a small-angle self-crossing in configuration space
which is a `2-encounter'; such configurations are called `Sieber-Richter pairs'
in the physics literature. Furthermore, we derive an estimate for the action
difference of the partners. In the second part of this paper [13], an inductive
argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit
Spectral statistics in chaotic systems with a point interaction
We consider quantum systems with a chaotic classical limit that are perturbed
by a point-like scatterer. The spectral form factor K(tau) for these systems is
evaluated semiclassically in terms of periodic and diffractive orbits. It is
shown for order tau^2 and tau^3 that off-diagonal contributions to the form
factor which involve diffractive orbits cancel exactly the diagonal
contributions from diffractive orbits, implying that the perturbation by the
scatterer does not change the spectral statistic. We further show that
parametric spectral statistics for these systems are universal for small
changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde
Classical orbit bifurcation and quantum interference in mesoscopic magnetoconductance
We study the magnetoconductance of electrons through a mesoscopic channel
with antidots. Through quantum interference effects, the conductance maxima as
functions of the magnetic field strength and the antidot radius (regulated by
the applied gate voltage) exhibit characteristic dislocations that have been
observed experimentally. Using the semiclassical periodic orbit theory, we
relate these dislocations directly to bifurcations of the leading classes of
periodic orbits.Comment: 4 pages, including 5 figures. Revised version with clarified
discussion and minor editorial change
The Mid-Pleistocene Transition induced by delayed feedback and bistability
The Mid-Pleistocene Transition, the shift from 41 kyr to 100 kyr
glacial-interglacial cycles that occurred roughly 1 Myr ago, is often
considered as a change in internal climate dynamics. Here we revisit the model
of Quaternary climate dynamics that was proposed by Saltzman and Maasch (1988).
We show that it is quantitatively similar to a scalar equation for the ice
dynamics only when combining the remaining components into a single delayed
feedback term. The delay is the sum of the internal times scales of ocean
transport and ice sheet dynamics, which is on the order of 10 kyr. We find
that, in the absence of astronomical forcing, the delayed feedback leads to
bistable behaviour, where stable large-amplitude oscillations of ice volume and
an equilibrium coexist over a large range of values for the delay. We then
apply astronomical forcing. We perform a systematic study to show how the
system response depends on the forcing amplitude. We find that over a wide
range of forcing amplitudes the forcing leads to a switch from small-scale
oscillations of 41 kyr to large-amplitude oscillations of roughly 100 kyr
without any change of other parameters. The transition in the forced model
consistently occurs near the time of the Mid-Pleistocene Transition as observed
in data records. This provides evidence that the MPT could have been primarily
a forcing-induced switch between attractors of the internal dynamics. Small
additional random disturbances make the forcing-induced transition near 800 kyr
BP even more robust. We also find that the forced system forgets its initial
history during the small-scale oscillations, in particular, nearby initial
conditions converge prior to transitioning. In contrast to this, in the regime
of large-amplitude oscillations, the oscillation phase is very sensitive to
random perturbations, which has a strong effect on the timing of the
deglaciation events
Semiclassical expansion of parametric correlation functions of the quantum time delay
We derive semiclassical periodic orbit expansions for a correlation function
of the Wigner time delay. We consider the Fourier transform of the two-point
correlation function, the form factor , that depends on the
number of open channels , a non-symmetry breaking parameter , and a
symmetry breaking parameter . Several terms in the Taylor expansion about
, which depend on all parameters, are shown to be identical to those
obtained from Random Matrix Theory.Comment: 21 pages, no figure
Semiclassical structure of chaotic resonance eigenfunctions
We study the resonance (or Gamow) eigenstates of open chaotic systems in the
semiclassical limit, distinguishing between left and right eigenstates of the
non-unitary quantum propagator, and also between short-lived and long-lived
states. The long-lived left (right) eigenstates are shown to concentrate as
on the forward (backward) trapped set of the classical dynamics.
The limit of a sequence of eigenstates is found
to exhibit a remarkably rich structure in phase space that depends on the
corresponding limiting decay rate. These results are illustrated for the open
baker map, for which the probability density in position space is observed to
have self-similarity properties.Comment: 4 pages, 4 figures; some minor corrections, some changes in
presentatio
Spectral Statistics of "Cellular" Billiards
For a bounded planar domain whose boundary contains a number of
flat pieces we consider a family of non-symmetric billiards
constructed by patching several copies of along 's. It is
demonstrated that the length spectrum of the periodic orbits in is
degenerate with the multiplicities determined by a matrix group . We study
the energy spectrum of the corresponding quantum billiard problem in
and show that it can be split in a number of uncorrelated subspectra
corresponding to a set of irreducible representations of . Assuming
that the classical dynamics in are chaotic, we derive a
semiclassical trace formula for each spectral component and show that their
energy level statistics are the same as in standard Random Matrix ensembles.
Depending on whether is real, pseudo-real or complex, the spectrum
has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types
of statistics, respectively.Comment: 18 pages, 4 figure
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